Theorem sge0tsms 47208

Description: Σ^{^} applied to a nonnegative function (its meaningful domain) is the same as the infinite group sum (that's always convergent, in this case). (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
sge0tsms.g	⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞))
sge0tsms.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
sge0tsms.f	⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))

Assertion

Ref	Expression
sge0tsms	⊢ (𝜑 → (Σ^‘𝐹) ∈ (𝐺 tsums 𝐹))

Proof of Theorem sge0tsms

Dummy variables 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2760	. . . 4 ⊢ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )
2	1	a1i 11	. . 3 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
3		xrltso 13192	. . . . . 6 ⊢ < Or ℝ*
4	3	supex 9434	. . . . 5 ⊢ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ V)
6		elsng 4598	. . . 4 ⊢ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ V → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )} ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )))
7	5, 6	syl 18	. . 3 ⊢ (𝜑 → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )} ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )))
8	2, 7	mpbird 260	. 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )})
9		sge0tsms.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
10	9	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
11		sge0tsms.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))
12	11	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
13		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
14	10, 12, 13	sge0pnfval 47201	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = +∞)
15	11	ffnd 6703	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝑋)
16	15	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹 Fn 𝑋)
17		fvelrnb 6938	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞))
18	16, 17	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (+∞ ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞))
19	13, 18	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞)
20		iccssxr 13483	. . . . . . . . . . . . . 14 ⊢ (0[,]+∞) ⊆ ℝ*
21		sge0tsms.g	. . . . . . . . . . . . . . 15 ⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞))
22		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ (𝒫 𝑋 ∩ Fin))
23	11	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,]+∞))
24		elinel1 4147	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
25		elpwi 4564	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
26	24, 25	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ⊆ 𝑋)
27	26	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ⊆ 𝑋)
28		fssres 6741	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 ⊆ 𝑋) → (𝐹 ↾ 𝑥):𝑥⟶(0[,]+∞))
29	23, 27, 28	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥):𝑥⟶(0[,]+∞))
30		elinel2 4148	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
31	30	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
32		0red 11235	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 0 ∈ ℝ)
33	29, 31, 32	fdmfifsupp 9345	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥) finSupp 0)
34	21, 22, 29, 33	gsumge0cl 47199	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ 𝑥)) ∈ (0[,]+∞))
35	20, 34	sselid 3929	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ 𝑥)) ∈ ℝ*)
36	35	ralrimiva 3154	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹 ↾ 𝑥)) ∈ ℝ*)
37	36	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹 ↾ 𝑥)) ∈ ℝ*)
38		eqid 2760	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥)))
39	38	rnmptss 7116	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹 ↾ 𝑥)) ∈ ℝ* → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ⊆ ℝ*)
40	37, 39	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ⊆ ℝ*)
41		snelpwi 5419	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ∈ 𝒫 𝑋)
42		snfi 9050	. . . . . . . . . . . . . . 15 ⊢ {𝑦} ∈ Fin
43	42	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ∈ Fin)
44	41, 43	elind 4146	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
45	44	3ad2ant2 1152	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
46	11	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝐹:𝑋⟶(0[,]+∞))
47		snssi 4746	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ⊆ 𝑋)
48	47	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → {𝑦} ⊆ 𝑋)
49	46, 48	fssresd 6742	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹 ↾ {𝑦}):{𝑦}⟶(0[,]+∞))
50	49	feqmptd 6946	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)))
51		fvres 6897	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ {𝑦} → ((𝐹 ↾ {𝑦})‘𝑥) = (𝐹‘𝑥))
52	51	mpteq2ia 5200	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)) = (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))
53	52	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)) = (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥)))
54	50, 53	eqtrd 2795	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥)))
55	54	oveq2d 7429	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐺 Σg (𝐹 ↾ {𝑦})) = (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))))
56	55	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐺 Σg (𝐹 ↾ {𝑦})) = (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))))
57		xrge0cmn 21657	. . . . . . . . . . . . . . . . 17 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd
58	21, 57	eqeltri 2856	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 ∈ CMnd
59		cmnmnd 19924	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
60	58, 59	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 𝐺 ∈ Mnd
61	60	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → 𝐺 ∈ Mnd)
62		simp2 1155	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → 𝑦 ∈ 𝑋)
63	11	ffvelcdmda 7077	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ (0[,]+∞))
64	63	3adant3 1150	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐹‘𝑦) ∈ (0[,]+∞))
65		dfss2 3917	. . . . . . . . . . . . . . . . . 18 ⊢ ((0[,]+∞) ⊆ ℝ* ↔ ((0[,]+∞) ∩ ℝ*) = (0[,]+∞))
66	20, 65	mpbi 233	. . . . . . . . . . . . . . . . 17 ⊢ ((0[,]+∞) ∩ ℝ*) = (0[,]+∞)
67	66	eqcomi 2769	. . . . . . . . . . . . . . . 16 ⊢ (0[,]+∞) = ((0[,]+∞) ∩ ℝ*)
68		ovex 7446	. . . . . . . . . . . . . . . . 17 ⊢ (0[,]+∞) ∈ V
69		xrsbas 17692	. . . . . . . . . . . . . . . . . 18 ⊢ ℝ* = (Base‘ℝ*𝑠)
70	21, 69	ressbas 17328	. . . . . . . . . . . . . . . . 17 ⊢ ((0[,]+∞) ∈ V → ((0[,]+∞) ∩ ℝ*) = (Base‘𝐺))
71	68, 70	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ((0[,]+∞) ∩ ℝ*) = (Base‘𝐺)
72	67, 71	eqtri 2783	. . . . . . . . . . . . . . 15 ⊢ (0[,]+∞) = (Base‘𝐺)
73		fveq2 6878	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
74	72, 73	gsumsn 20081	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) ∈ (0[,]+∞)) → (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))) = (𝐹‘𝑦))
75	61, 62, 64, 74	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))) = (𝐹‘𝑦))
76		simp3 1156	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐹‘𝑦) = +∞)
77	56, 75, 76	3eqtrrd 2800	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → +∞ = (𝐺 Σg (𝐹 ↾ {𝑦})))
78		reseq2 5967	. . . . . . . . . . . . . 14 ⊢ (𝑥 = {𝑦} → (𝐹 ↾ 𝑥) = (𝐹 ↾ {𝑦}))
79	78	oveq2d 7429	. . . . . . . . . . . . 13 ⊢ (𝑥 = {𝑦} → (𝐺 Σg (𝐹 ↾ 𝑥)) = (𝐺 Σg (𝐹 ↾ {𝑦})))
80	79	rspceeqv 3599	. . . . . . . . . . . 12 ⊢ (({𝑦} ∈ (𝒫 𝑋 ∩ Fin) ∧ +∞ = (𝐺 Σg (𝐹 ↾ {𝑦}))) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹 ↾ 𝑥)))
81	45, 77, 80	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹 ↾ 𝑥)))
82		pnfxr 11287	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ*
83	82	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ ℝ*)
84	38	elrnmpt 5942	. . . . . . . . . . . 12 ⊢ (+∞ ∈ ℝ* → (+∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹 ↾ 𝑥))))
85	83, 84	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (+∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹 ↾ 𝑥))))
86	81, 85	mpbird 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))))
87		supxrpnf 13370	. . . . . . . . . 10 ⊢ ((ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ⊆ ℝ* ∧ +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥)))) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)
88	40, 86, 87	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)
89	88	3exp 1137	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝑋 → ((𝐹‘𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)))
90	89	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (𝑦 ∈ 𝑋 → ((𝐹‘𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)))
91	90	rexlimdv 3161	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞))
92	19, 91	mpd 16	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)
93	14, 92	eqtr4d 2798	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
94	9	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
95	11	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
96		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
97	95, 96	fge0iccico 47198	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
98	94, 97	sge0reval 47200	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
99	23, 27	feqresmpt 6947	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥) = (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)))
100	99	adantlr 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥) = (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)))
101	100	oveq2d 7429	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ 𝑥)) = (𝐺 Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
102	21	fveq2i 6881	. . . . . . . . . . 11 ⊢ (+g‘𝐺) = (+g‘(ℝ*𝑠 ↾s (0[,]+∞)))
103		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞))
104		xrsadd 21603	. . . . . . . . . . . . . 14 ⊢ +𝑒 = (+g‘ℝ*𝑠)
105	103, 104	ressplusg 17376	. . . . . . . . . . . . 13 ⊢ ((0[,]+∞) ∈ V → +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))))
106	68, 105	ax-mp 5	. . . . . . . . . . . 12 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞)))
107	106	eqcomi 2769	. . . . . . . . . . 11 ⊢ (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) = +𝑒
108	102, 107	eqtr2i 2784	. . . . . . . . . 10 ⊢ +𝑒 = (+g‘𝐺)
109	21	oveq1i 7423	. . . . . . . . . . 11 ⊢ (𝐺 ↾s (0[,)+∞)) = ((ℝ*𝑠 ↾s (0[,]+∞)) ↾s (0[,)+∞))
110		icossicc 13489	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
111	68, 110	pm3.2i 476	. . . . . . . . . . . 12 ⊢ ((0[,]+∞) ∈ V ∧ (0[,)+∞) ⊆ (0[,]+∞))
112		ressabs 17340	. . . . . . . . . . . 12 ⊢ (((0[,]+∞) ∈ V ∧ (0[,)+∞) ⊆ (0[,]+∞)) → ((ℝ*𝑠 ↾s (0[,]+∞)) ↾s (0[,)+∞)) = (ℝ*𝑠 ↾s (0[,)+∞)))
113	111, 112	ax-mp 5	. . . . . . . . . . 11 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ↾s (0[,)+∞)) = (ℝ*𝑠 ↾s (0[,)+∞))
114	109, 113	eqtr2i 2784	. . . . . . . . . 10 ⊢ (ℝ*𝑠 ↾s (0[,)+∞)) = (𝐺 ↾s (0[,)+∞))
115	58	elexi 3472	. . . . . . . . . . 11 ⊢ 𝐺 ∈ V
116	115	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐺 ∈ V)
117		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ (𝒫 𝑋 ∩ Fin))
118	110	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (0[,)+∞) ⊆ (0[,]+∞))
119		0xr 11280	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ*
120	119	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 0 ∈ ℝ*)
121	82	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → +∞ ∈ ℝ*)
122	95	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝐹:𝑋⟶(0[,]+∞))
123	26	sselda 3931	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
124	123	adantll 727	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
125	122, 124	ffvelcdmd 7078	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,]+∞))
126	20, 125	sselid 3929	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℝ*)
127		iccgelb 13455	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹‘𝑦) ∈ (0[,]+∞)) → 0 ≤ (𝐹‘𝑦))
128	120, 121, 125, 127	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 0 ≤ (𝐹‘𝑦))
129		id 23	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑦) = +∞ → (𝐹‘𝑦) = +∞)
130	129	eqcomd 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑦) = +∞ → +∞ = (𝐹‘𝑦))
131	130	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → +∞ = (𝐹‘𝑦))
132	11	ffund 6707	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → Fun 𝐹)
133	132	ad2antrr 739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → Fun 𝐹)
134	22, 123	sylan 592	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
135	11	fdmd 6713	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → dom 𝐹 = 𝑋)
136	135	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑋 = dom 𝐹)
137	136	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑋 = dom 𝐹)
138	134, 137	eleqtrd 2862	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝐹)
139		fvelrn 7069	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ ran 𝐹)
140	133, 138, 139	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ran 𝐹)
141	140	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → (𝐹‘𝑦) ∈ ran 𝐹)
142	131, 141	eqeltrd 2860	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ ran 𝐹)
143	142	adantl3r 763	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ ran 𝐹)
144	96	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → ¬ +∞ ∈ ran 𝐹)
145	143, 144	pm2.65da 829	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑦) = +∞)
146	145	neqned 2962	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ≠ +∞)
147		ge0xrre 46361	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑦) ∈ (0[,]+∞) ∧ (𝐹‘𝑦) ≠ +∞) → (𝐹‘𝑦) ∈ ℝ)
148	125, 146, 147	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℝ)
149	148	ltpnfd 13172	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) < +∞)
150	120, 121, 126, 128, 149	elicod 13448	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,)+∞))
151		eqid 2760	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) = (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))
152	150, 151	fmptd 7107	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)):𝑥⟶(0[,)+∞))
153		0e0icopnf 13511	. . . . . . . . . . 11 ⊢ 0 ∈ (0[,)+∞)
154	153	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 0 ∈ (0[,)+∞))
155		eliccxr 13488	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
156		xaddlid 13294	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ* → (0 +𝑒 𝑦) = 𝑦)
157		xaddrid 13293	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ* → (𝑦 +𝑒 0) = 𝑦)
158	156, 157	jca 521	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ* → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
159	155, 158	syl 18	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,]+∞) → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
160	159	adantl 487	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ (0[,]+∞)) → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
161	72, 108, 114, 116, 117, 118, 152, 154, 160	gsumress 18784	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℝ*𝑠 ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
162		rege0subm 21636	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ∈ (SubMnd‘ℂfld)
163	162	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (0[,)+∞) ∈ (SubMnd‘ℂfld))
164		eqid 2760	. . . . . . . . . . . 12 ⊢ (ℂfld ↾s (0[,)+∞)) = (ℂfld ↾s (0[,)+∞))
165	117, 163, 152, 164	gsumsubm 18944	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℂfld ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
166		eqidd 2761	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℂfld ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℂfld ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
167		vex 3454	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
168	167	mptex 7222	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) ∈ V
169	168	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) ∈ V)
170		ovexd 7448	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld ↾s (0[,)+∞)) ∈ V)
171		ovexd 7448	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℝ*𝑠 ↾s (0[,)+∞)) ∈ V)
172		rge0ssre 13509	. . . . . . . . . . . . . . . . 17 ⊢ (0[,)+∞) ⊆ ℝ
173		ax-resscn 11181	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ⊆ ℂ
174	172, 173	sstri 3940	. . . . . . . . . . . . . . . 16 ⊢ (0[,)+∞) ⊆ ℂ
175		cnfldbas 21589	. . . . . . . . . . . . . . . . 17 ⊢ ℂ = (Base‘ℂfld)
176	164, 175	ressbas2 17330	. . . . . . . . . . . . . . . 16 ⊢ ((0[,)+∞) ⊆ ℂ → (0[,)+∞) = (Base‘(ℂfld ↾s (0[,)+∞))))
177	174, 176	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (0[,)+∞) = (Base‘(ℂfld ↾s (0[,)+∞)))
178	177	eqcomi 2769	. . . . . . . . . . . . . 14 ⊢ (Base‘(ℂfld ↾s (0[,)+∞))) = (0[,)+∞)
179	110, 20	sstri 3940	. . . . . . . . . . . . . . 15 ⊢ (0[,)+∞) ⊆ ℝ*
180		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (ℝ*𝑠 ↾s (0[,)+∞)) = (ℝ*𝑠 ↾s (0[,)+∞))
181	180, 69	ressbas2 17330	. . . . . . . . . . . . . . 15 ⊢ ((0[,)+∞) ⊆ ℝ* → (0[,)+∞) = (Base‘(ℝ*𝑠 ↾s (0[,)+∞))))
182	179, 181	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (0[,)+∞) = (Base‘(ℝ*𝑠 ↾s (0[,)+∞)))
183	178, 182	eqtri 2783	. . . . . . . . . . . . 13 ⊢ (Base‘(ℂfld ↾s (0[,)+∞))) = (Base‘(ℝ*𝑠 ↾s (0[,)+∞)))
184	183	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Base‘(ℂfld ↾s (0[,)+∞))) = (Base‘(ℝ*𝑠 ↾s (0[,)+∞))))
185		rge0srg 21651	. . . . . . . . . . . . . . 15 ⊢ (ℂfld ↾s (0[,)+∞)) ∈ SRing
186	185	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → (ℂfld ↾s (0[,)+∞)) ∈ SRing)
187		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → 𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
188		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
189		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (Base‘(ℂfld ↾s (0[,)+∞))) = (Base‘(ℂfld ↾s (0[,)+∞)))
190		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (+g‘(ℂfld ↾s (0[,)+∞))) = (+g‘(ℂfld ↾s (0[,)+∞)))
191	189, 190	srgacl 20344	. . . . . . . . . . . . . 14 ⊢ (((ℂfld ↾s (0[,)+∞)) ∈ SRing ∧ 𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
192	186, 187, 188, 191	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
193	192	adantl 487	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
194	172	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → (0[,)+∞) ⊆ ℝ)
195		id 23	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
196	195, 178	eleqtrdi 2870	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑠 ∈ (0[,)+∞))
197	194, 196	sseldd 3932	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑠 ∈ ℝ)
198	197	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → 𝑠 ∈ ℝ)
199	172	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → (0[,)+∞) ⊆ ℝ)
200		id 23	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
201	200, 178	eleqtrdi 2870	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑡 ∈ (0[,)+∞))
202	199, 201	sseldd 3932	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑡 ∈ ℝ)
203	202	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → 𝑡 ∈ ℝ)
204		rexadd 13284	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠 +𝑒 𝑡) = (𝑠 + 𝑡))
205	204	eqcomd 2766	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠 + 𝑡) = (𝑠 +𝑒 𝑡))
206	162	elexi 3472	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0[,)+∞) ∈ V
207		cnfldadd 21591	. . . . . . . . . . . . . . . . . . . . 21 ⊢ + = (+g‘ℂfld)
208	164, 207	ressplusg 17376	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0[,)+∞) ∈ V → + = (+g‘(ℂfld ↾s (0[,)+∞))))
209	206, 208	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ + = (+g‘(ℂfld ↾s (0[,)+∞)))
210	209, 207	eqtr3i 2785	. . . . . . . . . . . . . . . . . 18 ⊢ (+g‘(ℂfld ↾s (0[,)+∞))) = (+g‘ℂfld)
211	210, 207	eqtr4i 2786	. . . . . . . . . . . . . . . . 17 ⊢ (+g‘(ℂfld ↾s (0[,)+∞))) = +
212	211	oveqi 7426	. . . . . . . . . . . . . . . 16 ⊢ (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠 + 𝑡)
213	212	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠 + 𝑡))
214	180, 104	ressplusg 17376	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0[,)+∞) ∈ V → +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,)+∞))))
215	206, 214	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,)+∞)))
216	215	eqcomi 2769	. . . . . . . . . . . . . . . . 17 ⊢ (+g‘(ℝ*𝑠 ↾s (0[,)+∞))) = +𝑒
217	216	oveqi 7426	. . . . . . . . . . . . . . . 16 ⊢ (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡) = (𝑠 +𝑒 𝑡)
218	217	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡) = (𝑠 +𝑒 𝑡))
219	205, 213, 218	3eqtr4d 2805	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡))
220	198, 203, 219	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡))
221	220	adantl 487	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡))
222		funmpt 6571	. . . . . . . . . . . . 13 ⊢ Fun (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))
223	222	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Fun (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)))
224	150, 177	eleqtrdi 2870	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
225	224	ralrimiva 3154	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
226	151	rnmptss 7116	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → ran (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) ⊆ (Base‘(ℂfld ↾s (0[,)+∞))))
227	225, 226	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ran (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) ⊆ (Base‘(ℂfld ↾s (0[,)+∞))))
228	169, 170, 171, 184, 193, 221, 223, 227	gsumpropd2 18782	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℂfld ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℝ*𝑠 ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
229	165, 166, 228	3eqtrd 2799	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℝ*𝑠 ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
230	30	adantl 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
231	148	recnd 11261	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℂ)
232	230, 231	gsumfsum 21647	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
233	229, 232	eqtr3d 2797	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℝ*𝑠 ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
234	101, 161, 233	3eqtrrd 2800	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) = (𝐺 Σg (𝐹 ↾ 𝑥)))
235	234	mpteq2dva 5198	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))))
236	235	rneqd 5922	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) = ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))))
237	236	supeq1d 9416	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
238	98, 237	eqtrd 2795	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
239	93, 238	pm2.61dan 825	. . 3 ⊢ (𝜑 → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
240	21, 9, 11, 1	xrge0tsms 25061	. . 3 ⊢ (𝜑 → (𝐺 tsums 𝐹) = {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )})
241	239, 240	eleq12d 2854	. 2 ⊢ (𝜑 → ((Σ^‘𝐹) ∈ (𝐺 tsums 𝐹) ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )}))
242	8, 241	mpbird 260	1 ⊢ (𝜑 → (Σ^‘𝐹) ∈ (𝐺 tsums 𝐹))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  {csn 4584   class class class wbr 5103   ↦ cmpt 5186  dom cdm 5655  ran crn 5656   ↾ cres 5657  Fun wfun 6527   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413  Fincfn 8952  supcsup 9410  ℂcc 11122  ℝcr 11123  0cc0 11124   + caddc 11127  +∞cpnf 11264  ℝ^*cxr 11266   < clt 11267   ≤ cle 11268   +_𝑒 cxad 13161  [,)cico 13400  [,]cicc 13401  Σcsu 15773  Basecbs 17301   ↾_s cress 17322  +_gcplusg 17342   Σ_g cgsu 17525  ℝ^*_𝑠cxrs 17586  Mndcmnd 18836  SubMndcsubmnd 18890  CMndccmn 19907  SRingcsrg 20325  ℂ_fldccnfld 21585   tsums ctsu 24352  Σ^{^}csumge0 47190

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-inf2 9620  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202  ax-addf 11203  ax-mulf 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7678  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8159  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-map 8828  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-fsupp 9332  df-fi 9381  df-sup 9412  df-inf 9413  df-oi 9482  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-z 12616  df-dec 12737  df-uz 12888  df-q 12998  df-rp 13043  df-xadd 13164  df-ioo 13402  df-ioc 13403  df-ico 13404  df-icc 13405  df-fz 13562  df-fzo 13710  df-seq 14066  df-exp 14126  df-hash 14395  df-cj 15186  df-re 15187  df-im 15188  df-sqrt 15322  df-abs 15323  df-clim 15575  df-sum 15774  df-struct 17239  df-sets 17256  df-slot 17274  df-ndx 17286  df-base 17302  df-ress 17323  df-plusg 17355  df-mulr 17356  df-starv 17357  df-tset 17361  df-ple 17362  df-ds 17364  df-unif 17365  df-rest 17507  df-topn 17508  df-0g 17526  df-gsum 17527  df-topgen 17528  df-ordt 17587  df-xrs 17588  df-mre 17670  df-mrc 17671  df-acs 17673  df-ps 18654  df-tsr 18655  df-mgm 18730  df-sgrp 18821  df-mnd 18837  df-submnd 18892  df-grp 19060  df-minusg 19061  df-mulg 19191  df-cntz 19444  df-cmn 19909  df-abl 19910  df-mgp 20274  df-ur 20321  df-srg 20326  df-ring 20374  df-cring 20375  df-fbas 21582  df-fg 21583  df-cnfld 21586  df-top 23119  df-topon 23136  df-topsp 23158  df-bases 23171  df-ntr 23245  df-nei 23323  df-cn 23452  df-haus 23540  df-fil 24072  df-fm 24164  df-flim 24165  df-flf 24166  df-tsms 24353  df-sumge0 47191

This theorem is used by: (None)